Bð
ˆ
kÞ¼
i
2
½
¯
ð
2
P
þ
ð
ˆ
kÞ − ð
2
P
−
ð
ˆ
kÞ; ð8Þ
where we have introduced the so-called E and B polari-
zation modes. These modes offer an alternative description
of CMB linear polarization which, differently from Q- and
U-modes, is invariant under a rotation of the polarization
plane. In the following, we will use a description of the
radiation transfer both in terms of Q- and U-modes and
E- and B-modes.
The standard Boltzmann equations in the presence of
only linear scalar perturbations are given by [1]
d
dη
I
ðSÞ
þ iKμI
ðSÞ
þ 4½ψ
0
− iKμϕ
¼ τ
0
I
ðSÞ
− I
0ðSÞ
þ 4μv
B
−
1
2
P
2
ðμÞΠ
; ð9Þ
d
dη
P
ðSÞ
þ iKμP
ðSÞ
¼ τ
0
P
ðSÞ
þ
1
2
½1 − P
2
ðμÞΠ
; ð10Þ
d
dη
V
ðSÞ
þ iKμV
ðSÞ
¼ τ
0
V
ðSÞ
−
3
2
μV
1ðSÞ
; ð11Þ
where K denotes the Fourier conjugate of x and we are in a
coordinate system where Kk
ˆ
z axis. Here η is the conformal
time, the prime denotes differentiation with respect to
conformal time, ψ and ϕ are scalar cosmological (gravi-
tational potential) perturbations, v
B
is the electrons average
velocity, μ ¼
ˆ
k ·
ˆ
K ¼ cos θ, P
l
ðμÞ is the Legendre poly-
nomial of rank l and Π ≡ I
2ðSÞ
þ P
2ðSÞ
− P
0ðSÞ
, P being the
strength of the polarization field. The quantities I
lðSÞ
, P
lðSÞ
,
and V
lðSÞ
represent the lth order terms in the Legendre
polynomial expansion of the corresponding modes.
3
Finally, one defines the optical depth τðηÞ as
τ
0
ðηÞ¼−aðηÞn
B
x
e
σ
T
; τðηÞ¼−
Z
η
0
η
τ
0
ðη
0
Þdη
0
; ð13Þ
where n
B
is the electron density, x
e
is the ionization
fraction, and σ
T
is the Thomson cross section. For
more details about the derivation of Eqs. (9)–(11), see
Refs. [1,6,7].
4
In particular, it is possible to show that Eq. (10) admits
the general integral solution
P
ðSÞ
ðη
0
;K;μÞ¼
3
4
ð1 − μ
2
Þ
Z
η
0
0
dηe
iKðη−η
0
Þμ−τ
τ
0
Πðη;KÞ:
ð14Þ
Since scalar perturbations are invariant under rotations and
so axially symmetric around
ˆ
z, we get P
þ
¼ P
−
; thus,
U
ðSÞ
¼ 0 and scalar perturbations source only Q-modes.
Moreover, in this case (i.e., for scalar perturbations) the
spin raising and lowering operators act like (see e.g.,
Ref. [47])
¯
ð
2
P
ðSÞ
¼ ð
2
P
ðSÞ
¼ ∂
2
μ
½ð1 − μ
2
ÞP
ðSÞ
ðη
0
;K;μÞ: ð15Þ
Therefore, using the definitions (7) and (8), we get
E
ðSÞ
ðη
0
;K;μÞ¼−
3
4
Z
η
0
0
dηe
−τ
τ
0
Πðη;KÞ
× ∂
2
μ
½ð1 − μ
2
Þ
2
e
iKðη−η
0
Þμ
; ð16Þ
and
B
ðSÞ
ðη
0
;K;μÞ¼0: ð17Þ
This is the well-known result that linear scalar perturba-
tions cannot source B-mode polarization. In fact, it is well
known that the B-mode polarization in the CMB is
generated mainly by weak gravitational lensing and by
tensor perturbations [7,8]. Alternatively, a small amount of
B-modes can be generated also by second-order vector and
tensor modes sourced by scalar perturbations (see e.g.,
Refs. [48,49]).
Moreover, since in the right-hand side of Eq. (11) the
Stokes parameter V has no source terms in the case of
vanishing initial conditions for V, neither V-modes can arise
with linear scalar perturbations only.
From the next section, we will start to study the general
conditions for generating both V- and B-modes in the
presence of only linear scalar perturbations through their
direct coupling with E-modes generated by the photon-
fermion forward scattering. We will first write down a very
general form for the photon-fermion scattering amplitude
(Sec. III), and then we will apply it to the forward scattering
contribution in Eq. (2) (in Sec. IV).
III. GENERAL FORM OF THE PHOTON-FERMION
SCATTERING AMPLITUDE
We are interested in the Compton scattering of a photon
by a fermion (Fig. 1)
γðpÞþfðqÞ → γðp
0
Þþfðq
0
Þ; ð18Þ
3
We adopt the convention
I
lðSÞ
¼
Z
1
−1
dμ
0
2
Iðk; μ
0
ÞP
l
ðμ
0
Þ; ð12Þ
and an analogous expression for V
lðSÞ
and P
lðSÞ
.
4
We note that the Boltzmann equations do not coincide
between the different references. This is due to the fact that
each reference uses its own formal conventions. In our results,
we have followed the conventions of Ref. [1].
NICOLA BARTOLO et al. PHYS. REV. D 100, 043516 (2019)
043516-4